Quantile Regression Of Functional Data And Its Application

We extended multivariate quantile regression to functional data,and proposed a quantile regression model for where independent variables and response variable both are functional to estimate and conduct inference about the conditional quantile of response curve.In this model,we assumed the model error was a curve whose conditional quantile equaled to zero,and the regression coefficient curves at different quantile levels were obtained by using Bayesian estimation method or simplex method.A particular level of quantile determined the confidence interval for the dependent variable,and gave the distribution information of data which greatly improved the fitting effect of functional data and broadened the application scope of functional data analysis.For the functional data,the integral loss function was defined,finite nodes were selected for the data first,the parameter estimation values of the node were obtained by minimizing the loss function point by point,and the regression coefficient curves were obtained by smoothing the sequence of parameter estimation values.In the model,it was no longer assumed that the error was a Gaussian process with zero mean and homoscedasticity,which made up for the defects of the existing literature that the functional data linear model did not work well on data with skewed distribution or significant heteroscedasticity.As an application of the model,the logarithmic curve of precipitation was predicted based on the functional data of temperature curve.The curves with different quantile levels gave the distribution information of logarithmic curve of precipitation,which captured the short-term precipitation.Compared with the traditional functional data linear model,the prediction effect was obviously improved.